Existence and non-existence for time-dependent mean field games with strong aggregation

We investigate the existence of classical solutions to second-order quadratic Mean-Field Games systems with local and strongly decreasing couplings of the form $-\sigma m^\alpha$, $\alpha \ge 2/N$, where $m$ is the population density and $N$ is the dimension of the state space. We prove the existence of solutions under the assumption that $\sigma$ is small enough. For large $\sigma$, we show that existence may fail whenever the time horizon $T$ is large.


I
We consider in this paper systems of PDEs of the form where 0 is a smooth probability density, a smooth nal cost, is a bounded potential, and − is a monotone non-increasing coupling. As a model problem, we consider Such a system arises in the theory of Mean Field Games (MFG), a set of methods inspired by statistical physics to study Nash equilibria in (di erential) games with a population of in nitely many identical players. e MFG toolbox has been introduced in the mathematical community by the seminal papers [16,17,18], and by a series of lectures at Collège de France by P.-L. Lions [19]. A peculiarity of the present paper's se ing, is that the coupling ↦ → − ( ) is assumed to have a decreasing character in (the minus sign in front of is to emphasize this fact). Since − ( ) models the cost of a single agent in terms of the density of the population, (MFG) captures situations in which agents aim at maximizing aggregation. When the particular form (1) of is chosen, and are then related to the aggregation force. While (MFG) is known to enjoy uniqueness and long-time stability of solutions when ↦ → − ( ) is increasing (see e.g. [22] and references therein), the picture is less clear when ↦ → − ( ) is not increasing. Di erent phenomena have been observed in this framework, such as non-uniqueness of solutions [3,13], periodic solutions [7,11], and instability in the long-time horizon [20]. e main objective of this work is to investigate the existence of solutions when the coupling − has a strong decreasing character, that is when in (1) satis es e coe cient 2 turns out to be crucial if one looks at the variational side of the problem. e system (MFG) is indeed known to be the optimality conditions of a minimization problem with PDE constraints (or Mean Field type optimal control problem). Global minimizers of this problem have been shown to exist only if < 2 [13] (and these yield classical solutions [10]); when ≥ 2 , the variational problem is not even bounded from below. For this reason, the la er regime poses structural di culties even for the existence of solutions to (MFG). Solutions are indeed known to exist for general > 0 only when the time-horizon is small, by means of (nonvariational) techniques involving perturbations of the heat equation, see [2,9] (and references therein). We aim here at developing some new methods to explore existence without requiring to be "small enough".
Note that (2) implies ( ) ≥ +2 for some > 0. When has the form (1), then (2) holds for all > 0. Regarding the initial/ nal data and the potential , we assume that ∈ 2 (R ), and 2( − inf R ) + ∇ · ≥ 0 on R , ∈ 4 (R ), and ∇ · ≥ 0, 0 ∈ 4 (R ), 0 , | | 0 , 2 0 , ∇ 0 ∈ 1 (R ) and Note that the condition on is not much restrictive, and allows even for radially decreasing potentials (up to some degree). en we show that, if an additional condition involving 0 , , is satis ed, then (MFG) has no solutions if is large. eorem 1.1. Assume that (3), (4), (5) and (2) holds. Suppose that en, if Let us stress that the condition 0 > 0 may be realized or not depending on 0 and (and the oscillation of ). When ( ) = , note that for any xed 0 , replacing it by − 0 ( −1 ) into (6) yields hence 0 > 0 when the second term in the right-hans side is dominating, that is when is small enough or is large enough. In other words, non-existence is triggered by "concentration" of the initial datum, or "strength" of the aggregation force. e proof of eorem 1.1 involves the study of the evolution of second order moments ℎ( ) = ş 2 ( , ) . e core identity in Lemma 2.5 shows that under the standing assumptions, ℎ has to be strictly convex, but given the information at = 0, = , this forces ℎ to be negative when is large, which is impossible. Lemma 2.5 is based on two structural estimates: the rst one is the well-known conservation of energy (a quantity which stems from the Hamiltonian nature of (MFG)). e second one is a new identity which is obtained by testing the equations by projections of ∇ , ∇ over the direction , and is some sense related to dilations properties of the variational problem. We mention that a similar approach was used to obtain non-existence in [8] for stationary problems (for > 2 −2 ), but the analysis developed here for the evolutive case is more involved (and heavily related to the quadratic dependance with respect to the gradient in the rst equation, see Remark 2.7).
By the very same procedure, we obtain also non existence results for the so-called Planning Problem in MFG. In such a framework, one wants to drive agents from an initial con guration 0 to a nal one , optimizing some cost. is problem is related to a PDE system of the form (MFG), where there is no xed nal condition for , but rather a nal condition ( ) = . Our results on the planning problem are described in Section 2.1. eorem 1.1 leaves open the question, for a xed 0 , of the existence of solutions to (MFG) when is "small" (that is for small in the model case). Let us then describe the second main result of this paper. Assume that for some > 0, Note that we are not requiring to be increasing, but rather that grows at most like . e model case (1) perfectly falls into this se ing. We also suppose that en, we prove existence of solutions for small. (7) and (8). en, there exists 0 > 0 depending on , , 0 +1 (R ) , 2 (R ) , ||Δ || ∞ (R ) , such that for any ≤ 0 ( appearing in (7)), the system (MFG) has a classical solution ( , ). Note that if Δ = 0, then 0 is independent of .
We stress that 0 is a ected by only when the potential is non-trivial. When there is no spatial potential, i.e. ≡ 0, and is xed, solutions are proven to exist for all (and will probably "disappear" as → ∞, see Remark 3.3). Note also that we require rather smooth initial/ nal data, but existence restrictions depend only on 0 +1 (R ) and 2 (R ) (thus allowing to relax the smoothness assumptions via approximation arguments). As we previously observed, the only known existence results require small, and approach (MFG) as a perturbation of two heat equations; due to the presence of the non-linear term |∇ | 2 , this strategy does not allow to analyze the "small" regime. e key step here is an a priori estimate which is obtained heavily relying again on the MFG structure. We use a combination of the conservation of energy, socalled second-order estimates, and parabolic regularization to get an inequality of the form Note that in view of its super-linear nature, the previous estimate is meaningful only for small. In that case it is possible to set up a Schaefer's xed point procedure revolving around the boundedness of ť 2 +1 . Since < 2 −2 , this yields boundedness of ( ) in +2 2 , which is enough to set up a bootstrap procedure. We point out that the restriction 2 −2 on might be structural; we do not know at this stage how to construct solutions for ≥ 2 −2 and arbitrary . We nally mention that our existence scheme does not seem to apply easily to the Planning Problem. To our knowledge, when − is not increasing, existence is an open problem even in the short-time horizon regime.
Existence versus non-existence. For the sake of clarity, we summarize below, for xed initial-nal data 0 , and potential , existence and non-existence regimes as and vary. ese are sketched in Figure 1, for ≡ 0 and ≠ 0 (we again consider the model coupling (1)). First, we note that for any > 0, there exists = ( ) such that (MFG) has solutions provided that ≤ . ough this existence result is not stated explicitly anywhere, it can be derived via a straightforward adaptation of [13, eorem 1.4] from the at torus to the euclidean se ing R .
is (standard) short-time existence situation is light-blue coloured in Figure 1. Regarding our existence theorem, it says that there exists 0 = 0 ( ) such that (MFG) has solutions for all ≤ 0 . If ≡ 0, 0 is proven to be -independent, as shown in Figure 1 (green region).
Finally, we prove that (MFG) has no solutions whenever .
Equivalently, for all > 0 there exists * = * ( ) such that (MFG) has no solutions for any > * . is is the orange region in Figure 1. Note that the upper bound * ( ) on for which existence is expected goes to +∞ as → 0, while * ( ) → 2 Acknowledgements. e authors are members of the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). ey are partially supported by the research project "Nonlinear Partial Di erential Equations: Asymptotic Problems and Mean-Field Games" of the Fondazione CaRiPaRo.

N
is section is devoted to the proof of eorem 1.1, which will be based on several lemmas. Before we start, let us comment on the assumptions on , we will work with.
roughout the section we will assume that , ∇ , ∇ , Δ , , , ∇ , ∇ , Δ belong to (R × [0, ]). is degree of regularity is coherent with the one coming from the existence theorem that will be proven in the next section. Such a regularity can be obtained starting from any classical solution ( , ) ∈ 2,1 , by means of parabolic Schauder estimates (and the standing assumptions on 0 , , , ). All the arguments below actually need only polynomial growth in the -variable for , and their derivatives, and that | | 2 , ( ) ∈ 1 .
We stress that the assumption that ( , ) is a classical solution is not really crucial to get nonexistence. For example, arguing as in the proof of existence, a bootstrap procedure shows that weak solutions in a suitable (energy) sense have to be smooth, and thus eorem 1.1 applies. In other words, one can formulate the same non-existence result for a large class of weak solutions.
First, we show that if is bounded and smooth, it has to be a continuous ow of probability densities.
First, is non-negative by the maximum principle.
e following lemma describes the evolution of second order moments of , and concerns integrability properties of | ∇ | 2 and the crossed quantity |∇ ||∇ |.
Lemma 2.4. Let ( , ) be a classical solution of (MFG). In addition to the assumptions of previous Lemma 2.2, suppose that , ∇ , Δ , ∇ ∈ (R × [0, ]). en, the following statements hold: Proof. We start with claim (i). e (standard) idea is to multiply the second equation in (MFG) by and the rst one by , and perform several integration by parts. Since nothing is assumed regarding the integrability of , ∇ , . . . on R , we multiply the second equation in (MFG) by and the rst one by , where = − | | 2 2 , > 0, and integrate over R to get ż ], all the integrations by parts (using Lemma A.2 below) are justi ed. en, we will let → 0. We start with the rst two terms in (10), integrating repeatedly by parts to obtain ż us, plugging (11), (12) and (13) into (10) yields Again by the presence of , ∇ ∈ 1 (R ), and boundedness of , and their derivatives, we have and for all 1 ≤ 2 , Note now that en, since → 1 and ∇ , Δ → 0 uniformly on R as → 0, by the Dominated Convergence eorem one obtains for a.e. 1 , 2 ∈ (0, ). en, there exists ∈ R such that for a.e.
Note that if ∇ 0 ∈ 1 (R ), then ∇ (0)∇ 0 ∈ 1 (R ), and therefore the previous equality holds also for = 0. Now we prove claim (ii). By the Young's inequality, we have ż en by claim i), we obtain which is equivalent to ii). e next lemma is crucial. Exploiting the structure of the MFG system, it is possible to evaluate the second derivative in time of second order moments of in terms of integral quantities related to , and .
Proof. We multiply the second equation in (MFG) by · ∇ and the rst one by · ∇ , and integrate over × ( 1 , 2 ) to get We start with the rst two terms of the right-hand side. Integrating by parts, one has and therefore To handle the third and fourth term of the right-hand side of (14), the following formula will be useful 1 2 en, integrating by parts Regarding the last two terms in the right hand side of (14), by the de nition of and by integrating by parts we have We now manipulate the le -hand side of (14). We perform a rst integration by parts to have en, e equation for yields and therefore By plugging (15), (16), (17) and (18) into (14), we obtain Since , ∇ , ∈ ∞ (R × (0, )) and , |∇ | |∇ | ∈ 1 (R × (0, )), by Lemma A.1 we have en, since , ∇ · ∇ ∈ 1 (R × (0, )) and ∇ , , ∇ ∈ ∞ (R × (0, )), we obtain We now use Lemma 2.3 (i) to rewrite the le -hand side, and Lemma 2.4 (i) to replace the rst two terms of the right hand side to obtain for a.e. 1 , 2 1 2 en, dividing by 2 − 1 and taking the limit 2 → 1 we obtain the desired equality for a.e. . Note that ∈ ( 1 ) and is bounded, so ş R 2 ( ) agrees a.e. with a 1 function in .
We are now ready to prove eorem 1.1. We rst claim that ) we have ≥ 4 0 > 0. (22) en, by Lemma 2.5 and (22), we obtain ℎ ( ) ≥ 4 0 > 0 for all . Now we de ne ( ) = 2 0 2 + (2 − 4 0 ) + ℎ 0 and observe that en, by comparison, we derive can be "optimized". We exploit in particular the obvious fact that non-existence to (MFG) holds if and only if non-existence holds for the translated system we can conclude that the original MFG system (MFG) has no classical solutions whenever As a simple illustration, consider ≡ ≡ 0. It is clear that the quantity ↦ → ş R ( − ) 2 0 ( ) might be minimized by = 0, but this may be not the case for non-radially symmetric 0 .

Remark 2.7 ( e non-quadratic case). Consider a more general MFG system with power-like Hamiltonian
We observe that though the procedures described above yield meaningful identities also when ≠ 2, it is not clear how to conclude similar non-existence results. First, since the Hamiltonian nature of the MFG system is independent of > 1, one still has a conservation of energy of the form ż for all . Moreover, arguing as in Lemma 2.3, Finally, testing the equations by · ∇ and · ∇ respectively, and reasoning as in Lemma 2.5, one obtains where Roughly speaking, in a typical planning problem, one wants to drive the density of players from an initial con guration 0 to a target nal one . In [19], existence and uniqueness of smooth solutions to (MFG) is discussed. In [21], it is proven the existence of weak solutions when the Hamiltonian is not necessarily quadratic in ∇ . Note that both references consider the monotone case (− ( ) increasing) only.
We apply similar arguments as in the proof of eorem 1.1 to prove non existence for large time for the problem (28). for all ≥ 0. e main di erence is that we do not have any information on ( ), hence we cannot infer any information on the sign of ℎ ( ). On the contrary now , we obtain the desired contradiction, since ℎ has to be nonnegative. Note that one could choose in a way that it satis es ( ) = 4 0 and agrees with ℎ 0 and ℎ at time = 0 and = respectively, to improve when ℎ 0 ≠ ℎ (but we avoid writing the computations here for the sake of simplicity).

E
In this section we prove the existence eorem 1.2. Note rst that (7) implies We will use a generalization of the Schauder xed point theorem, that we recall here for completeness (see eorem 5.1 of [1]). en F has a xed point in .
In order to prove eorem 1.2, we will need the following crucial a priori estimate.
Step 3: "second order" estimates for the MFG system. Computing the Laplacian of the equation for yields Recall that , ∇ , and space derivatives of are bounded up to the fourth order, and ∈ ( 1 ). Hence, we multiply by and integrate by parts using Lemma A.2 to obtain Using the equation for in the le -hand term and plugging the previous inequalities implies We now control the term ş ∇ (0)∇ (0) . By Lemma 2.4 (i) and integration by parts we have Since the second and third term in the le -hand side of the previous equality are positive and since the last term in the right-hand side is negative, we get and back to (36) we obtain Step 4: conclusion. Finally, we adopt the following notation for simplicity: By plugging (37) into (35), we have and Young's inequality implies for someˆ> 0 depending on . Since 4 > 1, by a further application of Young's inequality we can absorb the term involving in the right-hand side into the le -hand side, to get the conclusion. Note that the constant in the statement of the theorem depends on || 0 || +1 (R ) , and on , , hence on , , || || 2 (R ) . Now we prove eorem 1.2. Our aim is to apply the xed-point eorem 3.1. First, we exploit the presence of a quadratic Hamiltonian to employ the standard Hopf-Cole change of variables = − /2 . In the unknowns , the MFG a system reads as system of coupled linear equations in divergence form We are going to prove the existence of a smooth couple , solving this linear system, which yields immediately a solution to (MFG).
We are now in the position to apply eorem 3.1 (with = , = 0), that gives the existence of a xed point = F ( ), i.e. a couple ( , ) solving the linear system in the classical sense. A classical solution to the MFG system can be then recovered via the reverse change of variables = −2 log . states, for any xed coupling su ciently "small", the existence of solutions to the MFG system for all > 0. is opens the way to the study of the long-time behavior of and . ough this analysis is beyond the scopes of this paper, the proof of eorem 1.2 suggests that should "vanish" as → ∞. Indeed, satis es for some > 0 that does not depend on . erefore, indicating that dissipates as → ∞ (and hence there is no ergodic behavior). Note also that stability of the 2 +1 -norm of as → ∞ is su cient to produce global bounds on ∞ and ∞ ; arguing as in [12], a possibly stronger "smallness" condition on the coupling (i.e. a smallness condition on ) guarantees then uniqueness of the couple ( , ) for all .

A
A. S We collect in this appendix some results that are used throughout the paper. We begin with some facts that are useful to integrate by parts on R .
en, the following equality holds: Note that an analogous formula holds on R with the Lebesgue measure .
Proof. e integration by parts formula holds on any bounded domain of the form By the previous Lemma A. 1 so that it su ces to pass to the limit → ∞ in the rst equality to obtain the assertion.
e following lemma is a (standard) regularity result for linear parabolic equations. Since we have not been able to nd it in this precise form in the literature, we sketch its proof for the reader's convenience.
Remark A.4. Note that a similar result applies to the backward equation More speci cally, under the same assumptions of Lemma A.3, we have the same estimates with ( ) in place of (0).
Finally, we recall some facts on distributional solutions (and their moments) to Fokker-Planck equations.